Real-rooted atlas

A038303 → basis A013609

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(11 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111213
2121444169
3133172613282197
4146411064829043521628561
516105114641053240968088032371293
617715611932612878460212960290402112644826809
7194871712480185413528284409948074536081312492812862748517

Show all 15 stored rows

Generated b-files: A001022 (row sums), A038217 (reversed triangle), A038316 ?

Computed in your browser ?

Technical details
Node id
n-10f70e3ea8b205cf · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-9bb94729b5d21425

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         11,          2;
  2:        121,         44,          4;
  3:       1331,        726,        132,          8;
  4:      14641,      10648,       2904,        352,        16;
  5:     161051,     146410,      53240,       9680,       880,       32;
  6:    1771561,    1932612,     878460,     212960,     29040,     2112,      64;
  7:   19487171,   24801854,   13528284,    4099480,    745360,    81312,    4928,    128;
  8:  214358881,  311794736,  198414832,   72150848,  16397920,  2385152,  216832,  11264,   256;
  9: 2357947691, 3858459858, 2806152624, 1190488992, 324678816, 59032512, 7155456, 557568, 25344, 512;
%F Formula
The row polynomials satisfy P(n,x) = (11+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(11 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (11 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038217: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,    11;
  2:   4,    44,    121;
  3:   8,   132,    726,    1331;
  4:  16,   352,   2904,   10648,    14641;
  5:  32,   880,   9680,   53240,   146410,    161051;
  6:  64,  2112,  29040,  212960,   878460,   1932612,    1771561;
  7: 128,  4928,  81312,  745360,  4099480,  13528284,   24801854,   19487171;
  8: 256, 11264, 216832, 2385152, 16397920,  72150848,  198414832,  311794736,  214358881;
  9: 512, 25344, 557568, 7155456, 59032512, 324678816, 1190488992, 2806152624, 3858459858, 2357947691;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(11 + 2*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (11 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038316: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,          2;
  2:        121,         44,          4;
  3:       1331,        726,        132,          8;
  4:      14641,      10648,       2904,        352,        16;
  5:     161051,     146410,      53240,       9680,       880,       32;
  6:    1771561,    1932612,     878460,     212960,     29040,     2112,      64;
  7:   19487171,   24801854,   13528284,    4099480,    745360,    81312,    4928,    128;
  8:  214358881,  311794736,  198414832,   72150848,  16397920,  2385152,  216832,  11264,   256;
  9: 2357947691, 3858459858, 2806152624, 1190488992, 324678816, 59032512, 7155456, 557568, 25344, 512;
%F Formula
The row polynomials satisfy P(n,x) = (11+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(11 + 2*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (11 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-71aa91d8d70f875b
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-71aa91d8d70f875b
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-71aa91d8d70f875b

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  4. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003